I FOr :lorD
IB·JGT·K·TUB I

STATISTICS

Paper II
ITime Allowed: Three Hours I Marks: 2001
INSTRUCTIONS
Candidates should attempt questions 1 and 5
which are compulsory, and any THREE ofthe

remaining questions, selecting at least ONE -question from each Section. All questions carry equal marks.
Marks allotted to each parl of a question are
indicated against each.
Assume suitable data, ifconsidered necessary,
and indicate the same clearly.
Answers must be written in ENGLISH only.

Unless otherwise indicated, symbols arid
notations have their usual meanings.

SECTION A . (Industrial Statistics and Techniques)
1. Attempt any four of the following: 4xlO==40
Explain briefly the dominance principle as
applied to game theory. Find the optimal strategies for both' players and the value of the game whose pay-off matrix is


(b)Between a single sampling plan with n 20, C 2 and a double sampling plan with n1 10, c1 n2 10, c2 can it be said that the second inspection scheme is more economical than the first Give reasons justifying your
answer.
rA study about a population showed that the mobility of population of a state to a village, town and city-is in the following percentages
Village Village Town City
60 25 15
Town 10 70 20
City 10 30 60

What will be the proportion of population in
village, town and city after one year and two
years, given that the" present population has
proportions of 0·40 and 0·10 in the village,
town and city respectively

Cd) "An engineer approximates the reliability of a system by

o t<t
o

"I a t t
o
Determine the failure rate. Does the failure rate increase or decrease with time?
.. 8-JGT-K-TUB 2 [Contd.] .
What is a multi-channel queuing problem Deduce differenti,al-difference equations for the (l1IMJc) (coIlfCFS) system.
2. Distinguish between process control and product controL Justify the use of three-sigma limits in the const:r:uction of control charts. Obtain, the control limits for p-chart when the size is fixed fo'r each sample. 12
.. Distinguish balanced and unbalanced

transportation problems. Solve the problem:
Destinations
5 3 7 3 8 5
5 6 12 5 7 11
Ports 2 8 3 8 2
9 6 10 5 10 9

Availability 3 3 6 2 1 2
Stating the underlying assumptions,
following 14
Requirement 3 4
2
8
fmd the
of the probability of happening of n events in the time interval in Poisson process.. Hence otherwise, obtain the distribution of inter-arrival times.
14
B-JGT-K-TUB" 3 [Contd.l
"I

3. Discuss the role of random numbers In simulation. Explain the basic steps of Monte-Carlo simulation. Let a six-faced homogeneous die be thrown. Let" N be the generated random number with the following rules for determining the outcome of the throw:
Outcome

1
2
3"
4 5
6
Rule

6
1

63 11

3 2

1 2

23 25

36 5

6
If the 10 random nUmbers '-drawn are 058, 6731
"
479, 948, 613, 593, 934, 178, 347 and 564"j thel1
"find the ,sequence of outcomes of ten throws. 13
What is the basic difference between
deterministic and probabilistic inventory
models Derive an" expression for finding
optimum inventory level in a single period
stochastic model without 'set-up cost with
continuous demand and discrete replenishment
unit. 15

B-JGT-K-TUB 4
(Cootd.l
Explain the need of censoring In life testing
giving suitable examples. In case of number of
failure censoring and time censoring, discuss
the estimation of parametef of exponential
distribution. 12
4. A system S consists of two subsystems A and
each having a reliability p. Subsystems A and B
are connected in series in the system. In order to
increase reliability of the system, it was decided.
to use four subsystems, the .possible combinations
being IB). The possible
configurations with these pairs are shown ·below.
Determine which configuration has the highest
reliability. 14







B-JGT-K-TUB [Contd.]
What is replacement problem? Derive condition for optimal replacement policy of capital equipment that deteriorates gradually.
For a machine, the following data are available:
Year o . 1 2 3 4 5
Cost of spares 200 400 700 1000 1400 -1600
Salary of maintenance staff 1200 1200 1400 1600 -2000 2600
Losses due to breakdown 600 800 700 1000 1200 1600
Resale value 12000 6000 3000 1500 800 400 '400

Determine the optimum period for replacement of the above machine.
What do you mean by transiti9n probability matrix (T.P.M.) of a Markov chain? Given the following T.P.M. of a Markov chain having three states 2 and with initial distribution

find the value of P[X3 Xl Xo
0·1 0·5 0·4
0·6 0·2 0·2
0·3 0·4 0·3 12

B-JGT-K-TUB 6 [Contd.]
SECTION B
(Quantitative Economics and Official Statistics)

5. Attempt any four of the following: 4xlO=40

Enumerate the items on which information was collected in 2001 population census of India.


Describe the link relative method for measuring seasonal variations in a time series.


.Prepare the life table upto age 3 starting with radix lo 100009 on the basis of the following values in thousands·



Cd) What is base shifting? From the index numbers given below, find the index numbers by shifting ·base year from 1976 to 1979
Year 1976 1977 1978
."
1979

1980
Index Number
100
110
175
250
300
What do you mean by autocorrelation? Discuss its consequences on OLS estimators in a linear. regression model.
7 [Contd.)


6. Show how would you study the cyclical component by harmonic analysis. For the autoregressive series
ut+2 aUt+1 Ct+2
show that if Et is a random variable and the
series is long then
var ut 1.

var c b)2 a2) .
t 14

What do you mean by stable and quasi-stable populations? What are the basic assumptions of stable population theory Derive three basic equations which provide-·information about the intrinsic growth rate, birth rate and age distribution of the population. 14


Write a note on the process of collection, compilation and publication of trade and p·rice statistics in India. 12


7. Starting from a sui.table, assumpti.on regarding. the relative growth rate of the population, derive the logistic equation. 14
What do you mean· by reliability of a test Compare the relative merits and demerits of various methods of finding reliability. Show that when the reliability coefficient IS zero, the standard error of an obtained score equals the standard deviation of the test. 14
f
What are index numbers? State their uses and limitations. Show'that the harmonic mean of the price relatives using current year's expenditure as weights the same result as obtained by
•
Paasche's formula. 12
B-JGT-K-TUB 8 lContd.]
8. Explain the meaning of multicollinearity of the
regression models with examples. P:resent a brief
account of the methods to overcome the problem
of multicollinearity. Describe a test for testing
multicollinearlty. 14
Explain crude and standardized death rates. In
what is standardized death rate superior "to
crude death rate? Discuss briefly the direct and
indirect methods of finding standardized death
rates. 14
. What is the concept of factor analysis? Mention
briefly the steps for performing factor analysis
and its application to Psychometry. 12


"B-JGT-K-TUB
9


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